Journal of Fish Biology (1998) 53, 237–258
Article No. jb980698
Stochastic simulation of size variation in turbot: possible
causes analysed with an individual-based model
A. K. I*‡, T. N†  A. F*
*Department of Fisheries and Marine Biology, University of Bergen, High Technology
Centre, N-5020 Bergen and †Department of Mathematics, University of Bergen,
N-5020 Bergen, Norway
(Received 1 September 1997, Accepted 20 March 1998)
Causes of size variation in a population of juvenile turbot were studied using an individualbased model (IBM). Each simulation started with 800 (divided into eight groups of 100 each)
120-day-old (posthatch) juveniles and was run for 140 days, and the data gained from model
simulations compared directly with the result of a laboratory study with size-graded turbot.
Stochastic growth with memory, which was included in the models as an individual genetical
growth rate variation, is important in explaining size variation, and the combination between
individual genetic growth rate and social interactions related to size-dependent hierarchies also
contributes to size variation. The use of size-dependent growth rate alone fails to explain size
variation, and is of little value in predicting size variation in turbot culture. Further, the results
indicate formation of different types of size hierarchies for different sizes of juvenile turbot.
1998 The Fisheries Society of the British Isles
Key words: growth; individual-based model; size hierarchy; genetic variation; juvenile turbot;
Scophthalmus maximus.
INTRODUCTION
Variation in individual growth rates can lead to high variation in sizes of individual
fishes of the same age, which is a typical feature seen in natural fish populations
(e.g. DeAngelis et al., 1993; Letcher et al., 1996) and in laboratory studies with
many fish species (e.g. Folkvord, 1991; Carmichael, 1994; Forsberg, 1996; Aune
et al., 1997) including turbot Scophthalmus maximus (Rafinesque) (Rosenberg &
Haugen, 1982; Nijhof, 1994; Imsland et al., 1995, 1996). From an ecological
viewpoint it is important to gain a better insight into the factors that govern these
size variations, as variations in growth and hence fish sizes within a population can
have a dramatic effect on the dynamics of a cohort (Chambers & Leggett, 1992;
DeAngelis et al., 1993; Rice et al., 1993). In an aquaculture situation such large
variation in sizes of similar aged fish has a negative impact on the production
control and the profitability of the fish farm, as there exist large price variations
between different sizes of fish (Lavens & Remmerswaal, 1994; Sutherland, 1997).
Many different factors have been put forward to explain observed size
variation in a cultured fish population; e.g. genetic differences producing size
variability (Huston & DeAngelis, 1987; Forsberg, 1995, 1996) and social
interactions (Brett, 1979; Jobling, 1982; Jobling & Koskela, 1996). Social
‡Author to whom correspondence should be addressed. Tel.: +47 55 58 44 71; fax: +47 55 58 44 50;
email: [email protected]
237
0022–1112/98/080237+22 $30.00/0
1998 The Fisheries Society of the British Isles
238
. .    .
interactions can lead to decreased growth in low ranking individuals (Wirtz,
1974; Jobling, 1985; Koebele, 1985; Jobling & Reinsnes, 1987; Huntingford
et al., 1990; McCarthy et al., 1992; Jobling & Koskela, 1996), and to the
establishment of a size hierarchy. The establishment of a size hierarchy may be
caused by direct competition for food (McCarthy et al., 1992; Jobling &
Koskela, 1996) or by other, less well-understood, social interactions (Jobling,
1982). Large differences in individual growth performance have been noted in
juvenile turbot (Nijhof, 1994; Imsland et al., 1996), as have stable size hierarchies
(Gaumet, 1994; Imsland et al., 1996, 1997b), and there is a need to optimize
production characteristics (Lavens & Remmerswaal, 1994). In order to achieve
more homogeneous growth one needs to control factors that contribute significantly to growth variation in turbot. A first step is to analyse which factors
contribute most to the observed growth variation. One way to do this is to
analyse possible causes with a model simulation of a population where individual
growth trajectories are followed and the causes of variation are analysed.
Earlier simulation studies with wild populations have highlighted the importance of better understanding of the mechanisms behind growth variation in fish
(e.g. DeAngelis et al., 1993; Letcher et al., 1996), as this will contribute to our
understanding of survival and fish population dynamics. In their study Letcher
et al. (1996) analysed variation in survival of larval fish with an individual-based
model and found that, of the intrinsic variables analysed, growth variation
explained 70% of variance in survival and had at least equally strong impact on
survival in the fish population as extrinsic variables (predator size and density,
prey density) did. Further, DeAngelis et al. (1993) showed that formulation of
individual growth rates that are to some extent correlated over time (i.e.
individual growth with memory) could cause dramatic changes in the dynamics
of a fish cohort causing the distribution of growth rates in the population to
broaden, which resulted in larger size variation in the population.
The present study designed an individual-based simulation model to analyse
the contribution of different growth governing factors to the observed variation
in juvenile weights reared under constant environmental conditions. Individualbased models (IBMs) have gained popularity in recent years (DeAngelis &
Gross, 1992; Judson, 1994; Letcher et al., 1996; Rose et al., 1996) as they
simulate a population as a collection of individual organisms each with its own
life history. The IBM model applied here is aimed at studying the effect of
size-dependent growth, individual genetic growth function and size hierarchydependent growth, as well as the combinations between these factors, on size
variation in turbot. Further, to study the effect of different initial size variation
each simulation was run using the initial weights (see below for details) from the
laboratory study of Sunde et al. (1998). In their study, juvenile turbot were size
graded into three size groups: small, medium and large, i.e. individuals from each
size group were grown independently from individuals of other size groups, and
additional fish were held in ungraded (control) groups. Each simulation started
with a total of 800 juveniles [divided into eight groups of 100 fish as in the Sunde
et al. (1998) experiment] and was run for the same period of time (140 days) as
the Sunde et al. (1998) study so that the data from the model simulations could
be compared directly with the results of the Sunde et al. (1998) study. In this
way, our model predictions could be tested with an existing independent data set.
     
239
Start
(a) Start with initial mean weight ±
S.D.
for
each grading group and replicate (Table I, see also Sunde
et al., 1998). Draw 100 individuals from the normal
distribution.
(b) Draw random growth function (x) from the normal
distribution. For model 0, 2a and 2b let x = 1 for all
individuals. For model 1, 3a and 3b let x ~ N (mean = 1.0,
S.D.
= 0.4
Juvenile size
All models. Size-dependent
growth rate for juvenile
turbot (Nijhof, 1994)
Next day
Model 2a, 2b, 3a and 3b.
Size hierarchicaldependent growth rate
Growth
Model 1, 3a and 3b
Every 2 weeks draw new
semi-random growth function
Use correlation between growth
at weekn and week n+2
Collection at day 0 and every 2
weeks until day 140
F. 1. Flow diagram of the individual-based growth model applied in the present study. The model
calculates growth rates for each juvenile every day of the simulation. Parameters for growth and
correlation of growth between weekn and weekn+2 are based on laboratory experiments with
juvenile turbot (Nijhof, 1994; Imsland et al., 1997a, b, c) and halibut (Hallaråker et al., 1995).
Each simulation started with 100 juveniles using initial mean weight.. of the four grading
groups in the study of Sunde et al. (1998).
MATERIALS AND METHODS
MODEL DESCRIPTION
General
The model simulated growth in juvenile turbot starting at the age of 4 months
(posthatch) and varied over 140 days. A constant temperature (19 C) was applied. Each
fish was defined by its weight, no fish died during the simulated growth trial, and the fish
were fed ad libitum. To simulate growth every fish passed through the growth simulation
model every day (Fig. 1). The model included three major state-specific growth factors of
juvenile turbot, namely size-dependent growth rate, individual genetic growth function
and size hierarchical-dependent growth. Each state-specific growth function depended on
the weight of each juvenile. These growth functions are described in detail below.
Function parameters were determined in two ways. Where available functions from
previously published experiments on juvenile turbot were used (size-dependent growth:
Nijhof, 1994; Imsland et al., 1995, 1996), whereas in some cases published data were
recalculated to arrive at function parameters (individual growth function with memory:
Gaumet, 1994; Hallaråker et al., 1995; Imsland et al., 1996, 1997b). Two different
. .    .
240
T I. Initial mean weight (g), standard deviation (..), and minimum and maximum
weight (g) in each group of size graded juvenile turbot used in the individual-based model
simulation of growth in turbot
Grading
group
Small
Medium
Large
Ungraded
Replicate
Mean weight
(g)
..
(g)
Minimum
(g)
Maximum
(g)
1
2
1
2
1
2
1
2
3·47
3·51
6·83
6·99
10·30
10·55
6·50
6·72
0·81
0·83
1·57
1·82
1·23
1·82
2·67
2·61
1·55
1·95
3·26
3·49
7·30
7·66
1·64
2·38
5·22
5·96
9·90
10·77
12·80
14·94
12·20
12·97
Each grading group consisted of two replicate tanks. See Sunde et al. (1998) for further details of the
grading groups.
variants of size hierarchy were tested based on previously published literature (Jobling,
1982, 1995; Huntingford et al., 1990; McCarthy et al., 1992; Jobling & Bardvik, 1994;
Jobling & Koskela, 1996).
To study the effect of different initial size variations each simulation was run using
initial weight.. from the laboratory study of Sunde et al. (1998). In their study
juvenile turbot reared at 19 C and fed ad libitum were size graded into three size groups
(mean initial size.., Table I): small (3·40·9 g), medium (7·01·7 g) and large
(10·51·4 g), and additional fish were held in ungraded (6·62·6 g) groups to give a
total of four experimental groups, each being replicated. Each simulation started with
800 [divided into eight groups as in the Sunde et al. (1998) experiment] 120-day-old
(posthatch) turbot juveniles and was run for the same period of time as the Sunde et al.
(1998) study (140 days) so that the data from the model simulations could be compared
directly with the results of the Sunde et al. (1998) study.
GROWTH
General growth model
The initial weight (W0 in g) of each individual fish was drawn from a normal
distribution, N(ì, ó) where ì is the mean weight and ó is the standard deviation, and
where initial ì and ó are taken from the study of Sunde et al. (1998) (Table I). The
growth of individual i at time t was modelled as:
Wi (t+1)=Wi (t) exp[Gi(t)(Ät)] Xi(t) Zi(t)
(1)
where Wi (t+1) is the weight of fish i at time (t+1); Wi (t) is the weight of fish i at time t;
Gi(t) is the size specific growth rate of individual i at time t; Ät is the length of the time
interval under study (here Ät=1 as the model simulates growth on a 1-day basis); Xi(t) is
an individual stochastic growth factor (see below); and Zi (t) is a size dependent
hierarchical growth factor (see below).
Size-dependent growth rate [Gi(t)]
In many fish species growth rates are related inversely to size (Pedersen & Jobling,
1989; Fonds et al., 1992; Rijnsdorp, 1993; Björnsson & Tryggvadóttir, 1996; Imsland et
al., 1996). To include this relationship in the growth model, the equation given by Nijhof
     
241
(1994) was used, which is based on calculations from a number of growth studies on
juvenile turbot in the size range 3–3000 g:
Gi (t)=0·1148Wi (t) 0·5
(2)
where Gi (t) and Wi (t) are the specific growth rate and weight, respectively, of individual
i at time t. In the model simulation, size dependent growth rate was calculated for each
fish on each day according to the weight of the individual fish on that day.
Individual stochastic growth function [Xi(t)]
Every individual was assigned an individual growth factor Xi that indicated the relative
growth rate of the individual i in relation to average growth rate in the population.
Further, Xi for each individual was time dependent, as every individual was assigned a
new Xi every 2nd week (see details below). As Xi is a continuous random variable the
central limit theorem assures that X will follow a normal distribution with E(X)=1 and
..(X)=known. The probability distribution function of X is thus given as:
To find the Xi for each individual the cumulative distribution function F(X) was
estimated:
A random number (r) is drawn from the uniform distribution on section [0, 1] and set
F(X)=r to calculate Xi. An estimation of ..(X) was used which was based both on the
available literature (Forsberg, 1995, 1996) and recalculations from earlier studies
(Hallaråker et al., 1995; Imsland et al., 1995, 1996, 1997b). The recalculations were done
using data on individually tagged fish so that individual growth can be tracked. An
individual growth factor and its variance based were recalculated on the relative growth
rate of the individual in relation to average growth rate in the population and the
variance of these growth rates. A total of 920 individual growth trajectories was analysed
and, based on these data, an estimate for ..(X) was found and accordingly ..(X) was
set=0·4 in the model simulations.
Correlation between growth factors in weekn and weekn+2
There are three possible scenarios in the correlation between individual growth rate in
time: deterministic growth rate, random growth rate and random growth rate with
memory. Deterministic growth would mean that, once chosen, the value of X for each
individual would not alter in the time period studied. One way to study if this is the case
or not is to study growth rates of tagged fish over a given time period. In Fig. 2 the
standardized growth (GS) rates of 20 individually tagged turbot were followed over a
period of 168 days (recalculated from Imsland et al., 1996). GS was found using the
formula:
where Gi (t) is growth of individual i at time t, G
z (t) is the average growth of all n
individuals at time t, and ..G(t) is the standard deviation of the 20 growth rates at time
t. As growth rates of the individuals cross extensively (Fig. 2), a deterministic growth rate
for each individual seems unlikely. As pointed out by DeAngelis et al. (1993), correlated
242
. .    .
F. 2. Individual standardized growth rate trajectories of 20 individually tagged turbot. Data are
recalculated from Imsland et al. (1996). See Materials and Methods for details of recalculations.
growth rates cause the distribution of growth rates in the population to broaden, whereas
the outcome of growth simulations with random individual growth rates without memory
is similar to the outcome of simulations with no random variation. To study the
correlation between growth rates in one period with the growth rate in an adjacent
period, Spearman’s rank correlation (rSP) coefficient was calculated from a number of
laboratory studies with turbot (Gaumet, 1994; Imsland et al., 1995, 1997a, b) and halibut
Hippoglossus hippoglossus (L.) (Hallaråker et al., 1995). In all of these studies the fish
were weighed every 2 weeks so that individual growth could be estimated for a number of
2-week periods. The correlation between individual growth rates in weekn and weekn+2
was then tested for all of these studies. There was an overall positive correlation between
adjacent (i.e. between weekn and weekn+2) growth rates (rSP =0·4). Therefore, this study
used a stochastic growth rate with memory, and a new growth factor (Xi) was assigned to
each individual every 2 weeks and this factor was positively correlated (rSP =0·4) with the
previous one.
Size hierarchy-dependent growth rate [Zi(t)]
The formation of size hierarchies in fish is generally found to have negative effects on
overall biomass gain, and leads to inconsistency of growth among groups and heterogeneity in growth among individuals within a group (Jobling, 1995; Jobling & Koskela,
1996). Two variants of size hierarchies [Zi (t)] were included in the model. Different
size hierarchy variants have been postulated based on experimental studies. One variant
was a dominance hierarchy in which every fish is subordinate to a larger individual
(McCarthy et al., 1992; Jobling, 1994). Another postulated variant of size-related
dominance was the division of a fish group into dominant (the largest fish) and
subordinates (the smallest fish) (Jobling, 1982; Huntingford et al., 1990). In both cases
the dominant fishes will eat a disproportionately high percentage of food provided
(Huntingford et al., 1990; McCarthy et al., 1992; Jobling & Bardvik, 1994). However,
although accepted to influence growth of farmed fish (Jobling, 1982, 1995; Huntingford
et al., 1990; McCarthy et al., 1992; Jobling & Koskela, 1996) it is difficult to obtain direct
information about the importance of social interactions on size variation in fishes. Thus
size hierarchy was included in our models to investigate to what extent the formation of
size hierarchy influences observed size variation. Two variants were tried (Za and Zb)
which simulate two postulated variants of size hierarchies in fishes (see above). These
variants simulate a size hierarchy in which every fish is subordinate to all fish that are
larger than that individual (Za), and a size hierarchy among the subordinate fish (Zb). As
the model results were compared with experimental results (Sunde et al., 1998), the same
experimental set-up was followed namely, eight groups of 100 fish in separate tanks.
     
243
Initially, several different modulation strengths of size hierarchy were tested, and based
on these initial trials, 0·1 was chosen as the difference between the highest and lowest
position.
(Za) decreasing sequence of 0·001 from 1·0 to 0·9 for all individuals in the same tank
(n=100 for each tank), i.e. for the largest fish at time t, Zi =1·0, for the second largest fish
at time t, Zi =0·999, the third largest fish at time t, Zi =0·998, . . ., the smallest fish at time
t, Zi =0·900.
(Zb) decreasing sequence of 0·002 from 1·0 to 0·9 for the 50 smallest (subordinates) fish
in every tank, i.e. Zi =1·0 for the dominants (n=50, the largest, second largest, . . ., fish
no. 51) at time t, for fish no. 50 at time t, Zi =0·998, for fish no. 49 at time t, Zi =0·996,
for fish no. 48 at time t, Zi =0·994, . . ., for fish no. 1 (the smallest fish) at time t, Zi =0·900.
MODEL TESTING AND APPLICATION
Six different versions of the general growth model (1) were tested in the present study.
Model 0: size-specific growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)]
(6)
Model 1: stochastic growth factor and size-specific growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)] Xi (t)
(7)
Model 2a: hierarchical growth factor Za and size-specific growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)] Zai (t)
(8)
Model 2b: hierarchical growth factor Zb and size specific growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)] Zbi (t)
(9)
Model 3a: hierarchical growth factor Za, stochastic growth factor, and size specific
growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)] Xi (t) Zai (t)
(10)
Model 3b: hierarchical growth factor Zb, stochastic growth factor, and size specific
growth
Wi (t+1)=Wi (t) exp[Gi (t)(Ät)] Xi (t) Zbi (t)
(11)
All models were tested using the statistical package S-PLUS (Venables & Ripley, 1994),
and every model testing with every 100 fish replicated in order to simulate replicated
tanks so that model simulations could be compared directly to the Sunde et al. (1998)
study.
DATA ANALYSIS AND STATISTICAL METHODS
A Kolmogorov–Smirnov goodness-of-fit two-sample test (Zar, 1984) was applied to
compare the weight distribution from each model simulation at every collection date
(every 2 weeks) with the corresponding data from laboratory studies (Sunde et al., 1998).
For each measurement the observed freqency was recorded. Then the cumulative
observed frequencies were determined from which the cumulative related frequencies
were obtained. Next for each measurement the corresponding expected frequency was
determined. In this context the expected frequency was set to be the laboratory data of
Sunde et al. (1998). Then the test statistic in this test was based on the largest disparity
between the observed and the expected data (Zar, 1984). Hence, if the Kolmogorov–
Smirnov test was not significant, the distributions tested were not different (Zar, 1984).
244
. .    .
For every weight distribution the skewness (ã1) of weight was calculated (Zar, 1984). The
Bonferroni correction (Johnson & Field, 1993) for the significance level (á) was applied
when comparing the different models at each collection date. Coefficient of variation
(CV) of weight for each replicate tank of the grading groups was regressed against time
(days) and analysed with linear regression (Zar, 1984). Separate regressions were made
for each model. The regression coefficients from these regressions were tested for
homogeneity with laboratory data using analysis of covariance (ANCOVA, Sokal &
Rohlf, 1995). In cases of homogeneity (i.e. parallel regression lines) a common regression
coefficient was calculated.
RESULTS
MODEL EVALUATION
The models were evaluated in two ways: (1) the weight distribution and
skewness gained from the different models at each collection date were compared
with laboratory data (Table II); and (2) the corresponding time pattern of CV of
the size–frequency distribution of the different models (Table III, Fig. 3) were
compared with laboratory data (Sunde et al., 1998).
WEIGHT DISTRIBUTIONS
The initial weight distribution of all models did not differ from the laboratory
data for all grading groups (Kolmogorov–Smirnov two sample test, P>0·7,
Table II). As the experiment proceeded there was, however, a large variation in
the fit of the different model data with corresponding laboratory data
(Kolmogorov–Smirnov two sample test, Table II). Overall, models 1, 3a and 3b
displayed the best fit to laboratory data (Table II). For the ungraded group,
models 3a and 3b were the only ones that displayed similar fits to those of the
corresponding laboratory data [Table II(d), Fig. 4].
The weight distribution gained by model 1 did not differ from that of the
laboratory data (Kolmogorov–Smirnov two sample test, P>0·05) from day 0
until day 28 for the small grading group [Table II(a)], from day 84 onwards
for the medium grading group (Table II(b)], and from day 56 to day 112 for the
large grading group [Table II(c)]. The weight distribution from model 3a did not
differ from corresponding laboratory data from day 84 onwards for the small
grading group [Table II(a)], from day 56 to 84 for the medium grading group
[Table II(b)], from day 28 to 56 for the large group [Table II(c)], and from day
42 to 70 for the ungraded group [Table II(d)].
Model 3b displayed the best fit of all models for the medium, large and
ungraded groups (Table II). For the small grading group, however, model 3a
displayed the best overall fit [Table II(a)]. The weight distribution from model
3b did not differ from corresponding laboratory data from day 126 onwards for
the small grading group [Table II(a)], from day 56 onwards for the medium
grading group [Table II(b)], from day 42 to 126 for the large group [Table II(c)],
and from day 70 to 126 for the ungraded group [Table II(d)].
Mean weight of the distributions gained by model 0 was similar to that of
the laboratory data, especially for the medium and ungraded groups [Table II(b)
and (d), respectively]. However, although displaying similar mean weights,
model 0 explained the observed size variation in the laboratory data poorly,
3·5 (0·8)
0·407
3·4 (0·9)*
0·060
3·5 (0·8)*
0·088
3·5 (0·8)*
0·303
3·5 (0·8)*
0·058
3·4 (0·7)*
0·185
3·5 (0·8)*
0·091
Day 0
5·8 (1·3)
0·050
7·1 (1·2)
0·270
7·2 (1·8)*
0·100
7·0 (1·3)
0·243
7·1 (1·3)
0·282
6·7 (1·9)
0·187
7·2 (2·1)
0·276
Day 14
8·6 (2·1)
0·183
12·1 (1·6)
0·199
12·2 (3·6)*
0·318
11·4 (1·9)
0·149
11·8 (2·0)
0·427
11·5 (3·8)
0·363
12·0 (4·1)
0·364
Day 28
13·1 (3·5)
0·142
18·3 (1·9)
0·157
18·5 (5·7)
0·446
16·8 (2·7)
0·061
17·6 (2·8)
0·529
17·6 (6·3)
0·406
18·1 (6·8)
0·438
Day 42
19·7 (5·5)
0·001
25·9 (2·7)
0·130
26·0 (8·4)
0·503
23·1 (3·7)
0·010
24·6 (3·9)
0·601
24·6 (9·3)
0·558
25·3 (9·9)
0·382
Day 56
26·9 (7·8)
0·009
34·7 (2·3)
0·111
34·9 (11·5)
0·507
30·1 (4·9)
0·068
32·6 (5·3)
0·649
32·7 (12·7)
0·640
33·8 (13·5)
0·367
Day 70
36·6 (11·8)
0·051
44·9 (3·1)
0·097
44·9 (14·4)
0·582
37·9 (6·5)*
0·115
41·6 (7·1)
0·681
41·6 (16·4)*
0·660
43·3 (17·4)
0·296
Day 84
47·7 (15·4)
0·180
56·4 (3·4)
0·085
56·5 (17·8)
0·714
46·4 (8·3)*
0·153
51·7 (9·1)
0·700
51·0 (19·9)*
0·580
54·0 (21·2)
0·192
Day 98
60·3 (20·3)
0·176
69·1 (3·8)
0·077
69·4 (21·4)
0·809
55·6 (10·5)
0·185
62·7 (11·6)
0·712
61·2 (23·9)*
0·552
65·7 (25·6)
0·146
Day 112
73·1 (25·4)
0·399
83·2 (4·2)
0·069
83·7 (25·0)
0·749
65·3 (13·0)
0·214
74·7 (14·5)
0·718
72·0 (28·7)*
0·572
78·5 (30·9)*
0·175
Day 126
88·3 (31·7)
0·722
98·5 (4·5)
0·063
99·1 (28·8)
0·572
75·6 (15·8)
0·239
87·6 (17·9)
0·721
83·6 (33·7)*
0·616
91·8 (36·5)*
0·199
Day 140
Results are given as mean (..); number of fish in each experimental category varied from 192 to 200 in each grading group, number of fish in each IBM was 200 at all times.
All results consist of two replicate tanks (n=100 in each replicate). Results of statistical analysis (Kolmogorov–Smirnov two-sample test, P<0·05) for comparison of the
sampling distribution of experimental data v. each model are given for each date. Model distributions significantly different from experimental data (P<0·05) are not marked.
Non-significant results are marked * and in italics. Skewness (ã1) of all weight distributions is also given.
Exp. data
ã1
Model 0
ã1
Model 1
ã1
Model 2a
ã1
Model 2b
ã1
Model 3a
ã1
Model 3b
ã1
Data
(a) Small group
T II. Mean weights (g) for all experimental categories (small, medium, large and ungraded) for both experimental data (see Sunde et al., 1998)
and all IBMs throughout the study
Day 0
Day 14
Day 28
Day 42
Day 56
Day 70
Day 84
Day 98
Day 112
Day 126
Day 140
Exp. data 6·9 (1·7) 10·4 (2·6) 15·4 (4·2) 21·6 (6·1) 30·9 (8·9) 40·3 (12·2) 52·4 (16·5) 66·1 (21·4) 82·0 (27·9) 97·8 (33·2) 119·0 (40·9)
ã1
0·211
0·404
0·631
0·656
0·564
0·681
0·595
0·518
0·530
0·531
0·603
Model 0 7·0 (1·6)* 11·9 (2·1) 18·1 (2·6) 25·6 (3·1) 34·4 (3·6) 44·5 (4·1) 55·9 (4·6) 68·7 (5·1) 82·7 (5·6) 98·0 (6·1) 114·5 (6·6)
ã1
0·179 0·248
0·292 0·321
0·343
0·359
0·371
0·382
0·390
0·397
0·403
Model 1 6·8 (1·7)* 11·6 (2·8) 18·1 (4·7) 26·1 (7·3) 35·2 (10·1) 45·5 (13·1) 56·9 (15·7)* 69·7 (18·4)* 84·4 (22·1)* 99·8 (25·9)* 116·2 (29·6)*
ã1
0·081
0·171
0·414
0·563
0·553
0·479
0·395
0·288
0·244
0·265
0·287
Model 2a 6·9 (1·9)* 11·4 (2·7) 16·7 (3·7) 23·0 (4·8)* 30·1 (6·2)* 37·8 (7·9)* 46·4 (9·8) 55·5 (12·1) 65·3 (14·6) 75·6 (17·6)
86·4 (20·9)
ã1
0·190 0·142 0·081 0·021
0·049
0·102
0·147
0·186
0·220
0·249
0·276
Model 2b 6·7 (1·6)* 11·3 (2·3) 17·1 (3·2) 23·9 (4·3) 31·9 (5·8) 40·8 (7·5)* 50·8 (9·7)* 61·8 (12·1) 73·7 (15·1) 86·5 (18·4) 100·3 (22·2)
ã1
0·097 0·102 0·262 0·387
0·479
0·547
0·594
0·627
0·649
0·664
0·673
Model 3a 7·1 (1·6)* 11·6 (3·1) 17·4 (5·5) 24·0 (8·2) 31·5 (12·0)* 40·0 (16·5)* 49·2 (20·7)* 59·3 (25·4) 69·9 (30·5) 81·2 (35·5)
93·0 (40·8)
ã1
0·009
0·563
0·584
0·568
0·667
0·794
0·891
0·867
0·783
0·763
0·764
Model 3b 7·1 (1·6)* 12·0 (3·1) 17·9 (5·5) 24·9 (8·3) 33·0 (11·4)* 42·1 (14·6)* 52·4 (18·5)* 64·2 (23·2)* 77·2 (28·5)* 90·9 (33·9)* 105·6 (40·0)*
ã1
0·266
0·340
0·426
0·352
0·282
0·164
0·084
0·106
0·188
0·222
0·179
Data
T II(b). Medium group
Day 0
Day 14
Day 28
Day 42
Day 56
Day 70
Day 84
Day 98
Day 112
Day 126
Day 140
Exp. data 10·4 (1·5) 14·6 (2·2) 20·8 (3·9) 28·6 (6·3) 39·9 (9·3) 51·6 (12·6) 68·9 (17·7) 85·0 (22·8) 103·1 (29·3) 121·8 (34·4) 146·4 (40·6)
ã1
0·147
0·388
0·378
0·561
0·359
0·141
0·099
0·101
0·143
0·332
0·312
Model 0 10·3 (1·3)* 16·2 (1·6) 23·3 (2·0) 31·8 (2·3) 41·5 (2·6) 52·6 (2·9) 64·9 (3·3) 78·5 (3·6) 93·5 (3·9) 109·7 (4·2) 127·2 (4·6)
ã1
0·160
0·120
0·091
0·072
0·057
0·047
0·037
0·030
0·024
0·019
0·014
Model 1 10·5 (1·2)* 16·6 (3·0) 24·3 (5·9) 32·9 (8·7) 42·7 (12·0)* 54·2 (15·2)* 67·0 (18·6)* 80·2 (21·6)* 94·6 (24·7)* 110·8 (28·0) 128·6 (31·3)
ã1
0·218
0·424
0·729
0·506
0·261
0·266
0·298
0·290
0·295
0·341
0·298
Model 2a 10·3 (1·3)* 15·4 (2·0) 21·5 (2·9)* 28·3 (4·0)* 36·0 (5·4) 44·3 (7·1) 53·2 (9·1) 62·8 (11·5) 72·9 (14·2) 83·5 (17·3) 94·6 (20·7)
ã1
0·063
0·003
0·057
0·099
0·133
0·163
0·189
0·212
0·235
0·256
0·276
Model 2b 10·4 (1·2)* 16·0 (1·9) 22·6 (2·8) 30·3 (4·0) 39·1 (5·6) 48·8 (7·5) 59·5 (9·8) 71·2 (12·6) 83·8 (15·7) 97·4 (19·4) 111·8 (23·5)
ã1
0·131 0·169 0·378
0·513
0·597
0·649
0·681
0·700
0·710
0·715
0·717
Model 3a 10·3 (1·2)* 16·2 (3·4) 22·7 (6·3)* 30·3 (9·5)* 38·5 (13·2)* 47·2 (17·3) 56·5 (21·1) 66·3 (25·0) 75·9 (29·2) 86·8 (33·9) 98·1 (39·1)
ã1
0·013
1·070
0·667
0·528
0·482
0·508
0·537
0·501
0·476
0·502
0·586
Model 3b 10·4 (1·3)* 16·1 (3·0) 23·0 (5·5) 31·1 (8·6)* 40·2 (12·0)* 50·1 (15·7)* 61·7 (20·4)* 73·8 (25·4)* 87·3 (30·6)* 102·2 (36·1)* 117·8 (41·6)
0·032
0·108
0·075
0·005
0·048
0·106
0·097
0·008
0·035
0·010
0·013
ã1
Data
T II(c). Large group
Exp. data
ã1
Model 0
ã1
Model 1
ã1
Model 2a
ã1
Model 2b
ã1
Model 3a
ã1
Model 3b
ã1
Data
6·6 (2·6)
0·386
6·6 (2·7)*
0·073
6·6 (2·5)*
0·090
6·9 (2·7)*
0·124
6·7 (2·5)*
0·289
6·6 (2·7)*
0·348
6·6 (2·7)*
0·178
Day 0
9·9 (3·5)
0·484
11·4 (3·6)
0·089
11·3 (3·9)
0·252
11·3 (3·7)
0·023
11·4 (3·5)
0·434
12·8 (5·4)
1·148
11·2 (4·1)
0·319
Day 14
14·7 (5·6)
1·015
17·4 (4·5)
0·189
17·4 (6·3)
0·332
16·7 (4·8)
0·015
17·2 (4·6)
0·515
16·8 (6·8)
0·508
17·1 (6·3)
0·224
Day 28
20·8 (8·0)
1·057
24·7 (5·3)
0·256
24·9 (8·9)
0·446
22·9 (6·2)
0·045
24·1 (6·0)
0·565
23·1 (9·3)*
0·608
23·9 (8·9)
0·264
Day 42
29·9 (11·8)
1·068
33·4 (6·2)
0·304
33·8 (11·7)
0·524
30·0 (7·7)
0·090
32·1 (7·7)
0·598
30·6 (12·4)*
0·738
32·0 (12·1)
0·401
Day 56
39·8 (15·5)
1·096
43·3 (7·1)
0·368
44·1 (15·1)
0·504
37·9 (9·5)
0·136
41·4 (9·6)
0·622
38·8 (15·8)*
0·860
41·6 (15·9)*
0·455
Day 70
52·0 (20·0)
0·987
54·6 (8·0)
0·391
55·7 (19·0)
0·359
46·4 (11·6)
0·179
51·1 (11·9)
0·636
47·6 (19·1)
0·860
51·9 (20·1)*
0·443
Day 84
T II(d). Ungraded group
65·8 (24·8)
0·981
67·1 (8·9)
0·409
68·5 (22·9)
0·134
55·6 (13·9)
0·218
62·1 (14·6)
0·652
56·5 (22·6)
0·773
63·7 (25·0)*
0·429
Day 98
81·1 (30·6)
0·984
80·9 (9·8)
0·424
82·1 (26·6)
0·265
65·4 (16·6)
0·253
74·1 (17·6)
0·660
66·1 (26·4)
0·764
76·2 (30·7)*
0·437
Day 112
95·3 (36·3)
0·980
96·1 (10·7)
0·437
97·4 (30·8)
0·252
75·7 (19·7)
0·284
87·0 (21·1)
0·665
76·1 (31·3)
0·746
90·1 (36·5)*
0·431
Day 126
116·6 (42·4)
1·102
112·5 (11·6)
0·340
114·1 (35·9)
0·339
86·6 (23·0)
0·311
100·9 (25·0)
0·668
87·1 (36·3)
0·789
104·9 (42·6)
0·403
Day 140
     
249
T III. Results from regression analysis of coefficient of variation (CV) for weight
regressed against time (days)
Grading group
Data
â
t(â)
P for â=0
Small
LD
M0
M1
M2a
M2b
M3a
M3b
LD
M0
M1
M2a
M2b
M3a
M3b
LD
M0
M1
M2a
M2b
M3a
M3b
LD
M0
M1
M2a
M2b
M3a
M3b
0·087
0·112
0·037
0·009
0·001
0·108
0·095
0·077
0·101
0·002
0·009
0·003
0·152
0·146
0·106
0·058
0·064
0·069
0·069
0·167
0·150
0·001
0·189
0·038
0·065
0·062
0·002
0·017
7·94
5·98
1·82
0·61
0·08
4·85
3·65
17·22
7·41
0·23
0·67
0·28
7·48
6·15
8·95
8·17
2·21
20·64
23·42
5·85
6·94
0·12
7·60
8·08
3·12
3·07
0·59
1·56
<0·01
<0·01
0·10
0·56
0·94
<0·01
<0·01
<0·01
<0·01
0·82
0·52
0·78
<0·01
<0·01
<0·01
<0·01
0·04
<0·01
<0·01
<0·01
<0·01
0·90
<0·01
<0·01
0·01
0·01
0·56
0·15
Medium
Large
Ungraded
LD, Laboratory data; M0, Model 0; M1=Model 1, etc. â is the regression coefficient in the regression
equation Y=á+âX where Y is CV and X is time. The t statistic and corresponding P value for the
hypothesis â=0 are individual.
and, apart from the initial weight distribution, model 0 always displayed a
significantly different weight distribution compared to the laboratory data
(Kolmogorov–Smirnov, P<0·01, Fig. 4).
Models 2a and 2b also displayed a poor fit to the laboratory data especially for
the large [Table II(c)], and ungraded [Table II(d), Fig. 4] groups.
On average, the laboratory data were skewed positively (i.e. skewed to the
right) around their mean value for all grading groups (Table II, Fig. 4). Models
1, 2a and 3a displayed the closest similarity to the distribution symmetry of the
laboratory data, as on average these models were also skewed positively for all
grading groups (Table II, Fig. 4). However, models 0 and 3b displayed in most
cases a symmetrical distribution about their mean value (Table II). The only
model displaying an average negative skewed distribution for all grading groups
was model 2b (Table II).
250
. .    .
F. 3. Coefficient of variation (CV) of weight plotted against time (days) for four different grading
groups: (a) small, (b) medium, (c) large, (d) ungraded. The result from laboratory data (Sunde
et al., 1998) and all simulation models run in the present study are shown. , Laboratory data;
, model 0; , model 1; , model 2a; , model 2b; , model 3a; , model 3b.
CAUSES OF SIZE VARIATIONS
For the laboratory data, the CV of weight increased in all grading groups
[linear regression, P<0·01, Table III, Fig. 3(a), (b), (c)] except the ungraded one,
where the CV was unaltered over time [P=0·9, Table III, Fig. 3(d)]. Models 3a
and 3b displayed the same pattern for CV as did the laboratory data although
these models had higher CV values for small, medium and large grading groups
[Fig. 3(a), (b), (c), respectively] compared with laboratory data. Further, models
3a and 3b had parallel regression lines with laboratory data for the small
(ANCOVA, F2,27 =0·26, P>0·75), large (F2,27 =2·02, P>0·15) and ungraded
(F2,27 =1·60, P>0·20) groups so that a common regression coefficient (â) could be
calculated. Hence, the CV increased with time for the small and large grading
groups for the laboratory data and models 3a and 3b (common â=0·097,
P<0·01, â=0·013, P<0·01, respectively), but was unaltered in the ungraded
group (common â=0·005, P>0·20).
For model 1, CV increased in all grading groups (Table III), but in contrast to
laboratory data CV decreased [P<0·01, Table III, Fig. 3(d)] over time for the
ungraded group. Model 1 and laboratory data had parallel regression lines
(F1,18 =1·84, P>0·15) for the large grading group (common â=0·085, P<0·01).
The CV values in models 2a and 2b always were lower than in the laboratory
data (Fig. 3). The time dependent pattern of CV was different between models
2a and 2b and laboratory data in all grading groups except for the large group,
where CV increased in both model and laboratory data [P<0·01, Table III,
     
Day 0
80
LD
60
60
40
40
20
20
0
0
80
M0
60
60
40
40
20
20
0
0
80
M1
60
60
40
40
20
20
Day 112
LD
M0
M1
0
0
No. of observations
251
80
M2a
60
60
40
40
20
20
0
M2a
0
80
M2b
60
60
40
40
20
20
0
M2b
0
80
M3a
60
60
40
40
20
20
M3a
0
0
80
M3b
60
60
40
40
20
20
0
0
5
10
15
20
0
25
0
Weight (g)
M3b
50 100 150 200 250
F. 4. Weight-frequency histogram for the ungraded group at two selected dates: day 0 and day 112. The
weight frequencies from laboratory data (Sunde et al., 1998) and all simulation models run in the
present study are shown.
Fig. 3(c)], whereas the regression lines were not parallel (ANCOVA, F2,27 =8·42,
P<0·01). In the other grading groups CVs for the models 2a and 2b data were
unaltered [small and medium groups, P>0·50, Fig. 3(a) and (b)], or decreased
with the passage of time [ungraded group, P<0·05, Fig. 3(d)].
252
. .    .
The CV for model 0 data decreased in all grading groups (P<0·01, Table III,
Fig. 3), and was always much lower during the latter part of the experiment than
found for corresponding laboratory data (Fig. 3).
DISCUSSION
THE MODEL APPROACH
To our knowledge this is the first published paper where individual-based
simulations were used to study fish growth in aquaculture. Earlier model
predictions in aquaculture have been based on modelling growth for a group of
fish, but not including day-to-day individual stochastic factors (Cuenco et al.,
1985; Forsberg, 1995; Rosa et al., 1997).
The major difference between previous IBM approaches and the one presented
in this paper is that our models are aimed at explaining size differences in culture
of a species, whereas earlier approaches have been aimed more generally at
explaining population dynamics in natural populations (e.g. Chambers &
Leggett, 1992; DeAngelis et al., 1993; Rice et al., 1993; Letcher et al., 1996; Rose
et al., 1996; Kristiansen & Svåsand, 1998). Another difference is the time scale
under study. The present paper studied growth of size graded turbot for a period
of 140 days during the juvenile stage with a simple simulation model including
only three growth governing factors and the combination between these factors.
In contrast modelling population dynamics of many marine fishes requires
long-term (multigenerational) simulations (Rose et al., 1996) where feeding,
growth, and mortality of every fish is simulated each day (DeAngelis et al., 1991;
Letcher et al., 1996; Rose et al., 1996) using detailed IBMs. The advantage of the
detailed model approach is a much greater flexibility in the questions that can be
addressed (Rose et al., 1996) whereas the main drawback can be the lack of
comparison between model simulations and experimental data. The present
study provided a realistic analysis of the findings by comparing directly the
results of our models with existing laboratory data (Sunde et al., 1998) giving a
realistic estimation and evaluation of our findings. DeAngelis (1988) pointed out
that perhaps the best approach in model simulations is to attempt to fractionate
the problem being studied into a large number of simpler ones that might be
addressed individually. The IBMs used in this paper are very simple and use
only a few equations derived mainly from existing literature (Gaumet, 1994;
Nijhof, 1994; Hallaråker et al., 1995; Imsland et al., 1995, 1996, 1997b), but
simulate events in the culture of turbot which are quite similar to the farming
situation. This is only the first step towards a better understanding of the factors
that govern size variation of cultured turbot. Important biotic variables such as
the metabolic cost of each individual (Nijhof, 1994), food consumption and
feeding efficiency (Devesa, 1994) are not included in the model applied here.
Later versions might include other growth governing factors, e.g. temperature
(Jones et al., 1981; Danielssen et al., 1990; Burel et al., 1996), photoperiod
(Fonds, 1979; Imsland et al., 1995, 1997c), salinity (Gaumet, 1994; Gaumet et
al., 1995), genotype specific growth (Nævdal et al., 1992; Torrissen & Shearer,
1992; Imsland et al., 1997a), and sexual dimorphism in growth (Devauchelle et
al., 1988; Déniel, 1990; Imsland et al., 1997b), and the duration of the study
     
253
might be extended to cover fully the period from early juvenile stage to sexual
maturation.
CAUSES OF SIZE VARIATION IN THE CULTURE OF TURBOT
Size variation in cultured turbot was related mainly to individual genetic
growth factors (model 1), and the combination of individual genetic growth
factors and social interactions related to size-dependent hierarchies (models 3a
and 3b). The use of size-dependent growth alone (model 0, Nijhof, 1994)
failed to explain the observed size variation (Figs 3 and 4) in all grading groups
(Tables II and III), and was of little value to predict size variation in turbot
culture. Further, our modelling efforts suggested that the individual genetical
variation in growth must be taken into account when designing future IBMs.
This is of vital importance in model studies of natural populations as growth
may have a large effect on survival (Letcher et al., 1996), and including
individual genetically based growth variations can increase the predicting power
of IBMs. Forsberg (1995, 1996) included a normally distributed genetical
growth index in his model assumptions, and found that his models followed
closely data gained from commercial-scale empirical results. For turbot, initial
trials (Gjerde et al., 1997) indicated that the heritability for body weight in turbot
is large, and that there is a substantial additive genetic variation in growth rate
in turbot which is promising for genetic improvement of the growth rate through
selective breeding.
The genetic growth factor alone (model 1) failed to explain observed size
variation in ungraded turbot [Tables II(d) and III], whereas the combination of
genetic growth factor and size-dependent hierarchy (models 3a and 3b) fitted
laboratory data well for ungraded turbot [Table III(d), Fig. 4]. This might
indicate that a size hierarchy influencing the growth of individual fish formed to
a larger extent in ungraded groups of turbot. Further, the increase in CV
observed in the present study amongst the small, medium and large groups
[Fig. 3(a), (b) and (c), respectively] might suggest some form of competition for
food or social interactions among juvenile turbot, because in populations where
the growth of some individuals was suppressed by food competition, CV
increased (Jobling, 1982). In some fish species dominant individuals exhibit
higher growth rates compared to subordinate individuals when reared together
(Brown, 1946; Magnuson, 1962; Jobling, 1985; Koebele, 1985). Different factors
have been proposed to explain why larger individuals suppress growth in smaller
individuals (Wirtz, 1974; Wallace et al., 1988), but food competition seems to be
particularly important (Magnuson, 1962; Wallace & Kolbeinshavn, 1988).
Further, Doyle & Talbot (1986) introduced the term resource-dependent competition, to describe the process where resource competition, e.g. for food,
controls the relative growth rates. The fish in the study of Sunde et al. (1998)
were size graded (Table I) to test the effect of different initial size variation on
subsequent growth, and to test whether the separation of small and large
individuals from each other resulted in less negative effects of social interactions
(Jobling, 1982, 1995; Knights, 1987) on growth. They found no negative effect of
size grading on growth and biomass gain, but pointed out that feeding every 6 min
and feeding in excess during those studies may have been important factors in
reducing aggression, so that the growth of the smallest individuals was not heavily
254
. .    .
suppressed by the larger individuals. This was indicated also in the study of
Jobling & Koskela (1996). They analysed interindividual variation in growth in
rainbow trout Oncorhynchus mykiss (Walbaum) during restricted feeding and in a
subsequent period of full feeding, and suggested that feeding hierarchies were
established under feed restriction but were broken down rapidly under full
feeding. However, there are cases in which size hierarchies develop in the presence
of excess food (Ehrlich et al., 1976; Jobling, 1982; McCarthy et al., 1992). In these
cases it is suggested that a size hierarchy (with larger fish dominating smaller ones)
forms where the dominant fish consumes a greater proportion of the group meal
(Huntingford et al., 1990; McCarthy et al., 1992; Jobling, 1995). It follows that,
although reduced under excess feeding, food competition and aggression might
still contribute to the formation of size hierarchies under such conditions. Our
findings support this as there are indications that size variation in turbot is partly
a consequence of size hierarchy formation.
There was a difference in the sensitivity of models 3a and 3b to the different
grading groups at different times, indicating formation of different types of size
hierarchies in different size groups. In the small grading group model 3a showed
the best fit to laboratory data [Table II(a)], whereas model 3b fitted the medium,
large and ungraded groups better than did model 3a [Table II(b), (c), and (d),
respectively]. This indicates a formation of size hierarchy in the medium, large
and ungraded groups where growth of the subordinates (here the 50 smallest fish)
was influenced negatively by the dominants, but whereas the size rank order
within the dominants (the 50 largest fish) was less important (factor Zb). In
contrast, the results indicate formation of a dominance hierarchy including all
the fish (factor Za) in the small grading group. Further, there was a shift from
factor Za towards factor Zb during the latter stages of the experiments (Table II)
for all grading groups, which might suggest formation of different types of size
hierarchies for different sizes of juvenile turbot.
CONCLUSIONS
The present modelling study indicates that there are two main causes for size
variation in laboratory studies with turbot:
(a) an individual genetical growth rate variation, which is stochastic in the
population and changes with time (stochastic growth with memory);
(b) the combination of individual genetical growth rate and social interactions
related to size-dependent hierarchies.
The use of size-dependent growth alone fails to explain size variation, and is of
little value in predicting size variation in turbot culture.
The work described in this paper was partly supported (AKI) by the Nordic Academy
for Advanced Study (NorFA). The authors thank L. M. Sunde and O
u . D. B. Jónsdóttir
for skilled technical assistance and two anonymous referees for valuable comments on the
manuscript.
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